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At the end of Inheritance, Eragon, in spite of himself,

leaves the charted lands of Alagaësia for the far east, where he'd be able to train future Knights far from peoples, kings and influences. Not even the deserts between the Elves and the Dwarfs are good enough, given concerns for the latters' sheep and whatnot. No, only the far east would be far away enough from everything. At least, that's what he says, and the book seems to take this for granted. However, that doesn't actually mean those lands are actually devoid of civilization.

Eragon would hardly have considered the trip, and the forever-and-ever consequences, without being positive about this.

How does he know for sure?

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Eragon says that the Eldunari of the very, very old dragons told him the lands to the far east were uninhabited, and that they had no reason to believe that those lands had become settled since they last checked. This occurs when he's telling Jeod about their decision to leave. From pages 814-815, U.S. hardcover:

“But you … you’re going to travel where few but the dragons or Riders have ever gone. Tell me, do you know what lies to the east? Is there another sea?”
“If you travel far enough.”
“And before that?”
Eragon shrugged. “Empty land for the most part, or so the Eldunarí say, and I have no reason to think that’s changed in the past century.”

As to how the Eldunari still know, it's mentioned earlier in the book that the those hidden on Vroengard had been watching the events of Alagaësia very carefully while cooped up in the Vault of Souls. This is Umaroth speaking:

"Together our strength is great, and we have been able to observe much of what has transpired throughout the land in the years since."

4

He didn't

How did Eragon even know that the far east was uninhabited?
No, he doesn't know if the far east is uninhabited.
Reddit AMA

And as it turned out, they were inhabited.

Among them lived groups of wandering tribes: strange, half-wild humans the likes of which Eragon had never encountered before. So far they had proven no trouble, but he remained wary.

The Fork, the Witch, and the Worm: Tales from Alagaësia (Volume 1: Eragon) - Chapter I - "Mount Arngor"

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